Frequency tracking for OFDM transmission over frequency selective channels

ABSTRACT

A system comprises a receiver module that generates a receiver carrier frequency and that demodulates an orthogonal frequency division multiplexing (OFDM) signal using the receiver carrier frequency. A CFO estimator module communicates with the receiver module and generates a carrier frequency offset (CFO) estimate using an M-th power method and normalizing a magnitude thereof. Another CFO estimator module communicates with the receiver module and generates a carrier frequency offset (CFO) of the OFDM signal using an M-th power method and normalizing a magnitude thereof. The CFO estimator module may be blind. The OFDM signal may also contain pilot subcarriers. The pilot subcarriers may contain known pilot symbols.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Nos.60/550,666, filed on Mar. 5, 2004 and 60/585,961, filed on Jul. 6, 2004.This application relates to Ser. No. 10/986,082 filed on Nov. 10, 2004,Ser. No. 10/986,110, filed on Nov. 10, 2004 and Ser. No. 10/986,130filed on Nov. 10, 2004. The disclosures of the above applications areincorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates to receivers, and more particularly toorthogonal frequency division multiplexing (OFDM) receivers.

BACKGROUND OF THE INVENTION

Orthogonal frequency division multiplexing (OFDM) has been used in bothwired and wireless communication systems because of its simpleimplementation and effectiveness in combating inter-symbol interference(ISI). However, OFDM is susceptible to carrier frequency offset (CFO).Even small frequency offsets can cause large signal-to-noise ratio (SNR)and bit-error-rate (BER) degradation. In particular, OFDM systemsemploying time-domain differential demodulation are very sensitive tothe CFO. Therefore, an accurate CFO estimation and correction algorithmsshould be employed to avoid performance degradation.

The frequency synchronization process can usually be split into anacquisition phase and a tracking phase. The CFO is estimated coarselyand quickly during an acquisition phase, and then a residual or smallCFO is estimated more accurately during a tracking phase.

Various CFO correction algorithms have been proposed for OFDM systems.In one approach, a cyclic prefix-based (CPB) algorithm was proposed. TheCPB algorithm does not require training symbols or pilot tones. However,the CPB algorithm does not perform well in frequency selective channelssince it was designed for an additive white Gaussian noise (AWGN)channel. On the other hand, a pilot tone-aided (PTA) algorithm canestimate the CFO more accurately in frequency selective channels.However, the PTA algorithm requires pilot subcarriers embedded among thedata subcarriers, which reduces available bandwidth.

SUMMARY OF THE INVENTION

A system comprises a receiver module that generates a receiver carrierfrequency and that demodulates an orthogonal frequency divisionmultiplexing (OFDM) signal using the receiver carrier frequency. A blindCFO estimator module communicates with the receiver module and generatesa carrier frequency offset (CFO) estimate of said OFDM signal using anM-th power method and normalizing a magnitude of said CFO estimate.

In some implementations, the CFO estimator module generates said CFOestimate based on

${\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\measuredangle\left( {\sum\limits_{k = 0}^{N - 1}\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} \right)}}},$wherein M is the number of phases, λ is a normalizing power, N is thenumber of data samples, α is a ratio of the number of cyclic prefixsamples divided by N, m is a symbol index, and Y_(m-1)*[k]Y_(m)[k] areadjacent received symbols.

In some implementations, the receiver module operates in afrequency-selective channel, an additive white Gaussian noise (AWGN)channel or a multi-path fading channel. The receiver module is awireless or wired receiver. The receiver module performs demodulationusing amplitude phase shift keying (APSK) or M-ary phase shift keying(MPSK). The OFDM signal contains pilot subcarriers. The pilotsubcarriers contain known pilot symbols.

In some implementations, the estimator module generates a first CFOestimate

estimate  ɛ̂_(_1),multiplies a time domain signal by

𝕖^(−j2πɛ̂_(_1)n/N)to generate an adjusted time domain signal, and uses the adjusted timedomain signal to generate a second CFO

estimate  ɛ̂_(_2).The estimator module multiplies the adjusted time domain signal by

𝕖^(−j2πɛ̂_(_2)n/N)to generate a second adjusted time domain signal, and uses the secondadjusted time domain signal to generate a third CFO

estimate  ɛ̂_(_3).

A system comprises a receiver module that generates a receiver carrierfrequency and that demodulates an orthogonal frequency divisionmultiplexing (OFDM) signal using the receiver carrier frequency. A CFOestimator module communicates with the receiver module and generates acarrier frequency offset (CFO) estimate of said OFDM signal using anM-th power method and normalizing a magnitude of said CFO estimate. TheOFDM signal contains pilot subcarriers.

In some implementations, the CFO estimator module generates said CFOestimate based on:

$\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {{\sum\limits_{k \in D}\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} + \frac{\left( {\sum\limits_{k \in P}{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}} \right)^{M}}{{{\sum\limits_{k \in P}{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*\;}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}}^{\lambda}}} \right)}}$wherein M is the number of phases, λ is a normalizing power, N is thenumber of data samples, α is a ratio of the number of cyclic prefixsamples divided by N, m is a symbol index, Y_(m-1)*[k]Y_(m)[k] areadjacent received symbols and X_(m-1)*[k]X_(m)[k] are pilot subcarriers.

In some implementations, the receiver module operates in afrequency-selective channel, an additive white Gaussian noise (AWGN)channel, or a multi-path fading channel. The receiver module is awireless or wired receiver. The receiver module performs demodulationusing amplitude phase shift keying (APSK) or M-ary phase shift keying(MPSK). The pilot subcarriers contain known pilot symbols.

In some implementations, the estimator module generates a first CFO

estimate  ɛ̂_(_1),multiplies a time domain signal by

𝕖^(−j2πɛ̂_(_1)n/N)to generate an adjusted time domain signal, and uses the adjusted timedomain signal to generate a second CFO estimate

estimate  ɛ̂_(_2).The estimator module multiplies the adjusted time domain signal by

𝕖^(−j2πɛ̂_(_2)n/N)to generate a second adjusted time domain signal, and uses the secondadjusted time domain signal to generate a third CFO estimate

estimate  ɛ̂_(_3).

Further areas of applicability of the present invention will becomeapparent from the detailed description provided hereinafter. It shouldbe understood that the detailed description and specific examples, whileindicating the preferred embodiment of the invention, are intended forpurposes of illustration only and are not intended to limit the scope ofthe invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will become more fully understood from thedetailed description and the accompanying drawings, wherein:

FIG. 1 is a functional block diagram of a baseband equivalent model ofan OFDM system with carrier frequency offset (CFO);

FIG. 2 is a graph of mean-square error (MSE) of a blind CFO estimationalgorithm for the AWGN channel with different values of λ in (17) whenQPSK is employed;

FIG. 3 is a graph of MSE of the blind CFO estimation algorithm for amultipath channel with different values of λ in (17) when QPSK isemployed;

FIG. 4 is a graph of MSE of the blind CFO estimation algorithm withdifferent number of iterations of the estimation and correction steps;

FIG. 5 is a graph of mean-square error of the cyclic prefix-based (CPB),pilot tone-aided (PTA), and blind algorithms;

FIG. 6 is a graph of average bit error rate for four-phase time-domaindifferential modulation with the CFO corrected by the cyclicprefix-based (CPB), pilot tone-aided (PTA), and blind algorithms and areference BER curve in the absence of the CFO;

FIG. 7 is a functional block diagram of an exemplary receiver thatestimates CFO and generates a CFO correction; and

FIG. 8 is a flowchart illustrating exemplary steps that are performed bythe receiver in FIG. 7.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The following description of the preferred embodiment(s) is merelyexemplary in nature and is in no way intended to limit the invention,its application, or uses. For purposes of clarity, the same referencenumbers will be used in the drawings to identify similar elements. Asused herein, the term module refers to an application specificintegrated circuit (ASIC), an electronic circuit, a processor (shared,dedicated, or group) and memory that execute one or more software orfirmware programs, a combinational logic circuit, and/or other suitablecomponents that provide the described functionality.

The present invention relates to carrier frequency offset (CFO)estimation for OFDM systems. CFO estimation according to the presentinvention provides improved performance as compared to conventional CFOestimation algorithms. For example, cyclic prefix-based (CPB) estimatorsare blind but do not perform well in frequency selective channels. Pilottone-aided (PTA) estimators perform well in frequency selective channelsbut require pilot subcarriers. Unlike the CPB and PTA estimators, theCFO estimator according to the present invention does not requiretraining symbols or pilot subcarriers and performs well in frequencyselective channels.

An OFDM system transmits information as a series of OFDM symbols.Referring now to FIG. 1, a baseband equivalent model of an OFDM systemis shown. As is shown in FIG. 1, the inverse discrete Fourier transform(IDFT) is performed on the information symbols X_(m)[k] for k=0, 1, . .. , N−1 to produce the time-domain samples x_(m)[n] of the m-th OFDMsymbol:

$\begin{matrix}{{x_{m}\lbrack n\rbrack} = \left\{ \begin{matrix}{{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{{X_{m}\lbrack k\rbrack}{\mathbb{e}}^{{j2\pi}\;{{k{({n - N_{g}})}}/N}}}}},{{{if}\mspace{14mu} 0} \leq n \leq {N + N_{g} - 1}}} \\{0,{otherwise},}\end{matrix} \right.} & (1)\end{matrix}$where N and N_(g) are the numbers of data samples and cyclic prefixsamples, respectively.

The OFDM symbol x_(m)[n] is transmitted through a channel h_(m)[n] andis corrupted by Gaussian noise {tilde over (z)}_(m)[n] The channelh_(m)[n] is assumed to be block-stationary, i.e., time-invariant duringeach OFDM symbol. With this assumption, the output {tilde over(y)}_(m)[n] of the channel can be found using a convolution operation asfollows:{tilde over (y)} _(m) [n]=h _(m) [n]*x _(m) [n]+{tilde over (z)} _(m)[n],  (2)where * denotes the convolution operation, i.e.,h_(m)[n]*x_(m)=[n]=Σ_(r=˜∞) ^(∞)h_(m)[r]x_(m)[n−r], and {tilde over(z)}_(m)[n] is additive white Gaussian noise with varianceτ_({tilde over (Z)}) ².

When the receiver oscillator is not perfectly matched to the transmitteroscillator, there can be CFO Δf=f_(t)−f_(r) between the transmittercarrier frequency f_(t) and the receiver carrier frequency f_(r). Inaddition, there may be a phase offset θ₀ between the transmitter and thereceiver carrier. The received symbol y_(m)[n] is theny _(m) [n]=e ^(j[2πΔf(n+m(N+N) ^(g) ^())T+θ) ⁰ ^(])(h _(m) [n]*x _(m)[n]+{tilde over (z)} _(m) [n]),  (3)where T is the sampling period.

The frequency offset Δf can be represented with respect to thesubcarrier bandwidth 1/NT by defining the relative frequency offset ε as

$\begin{matrix}{ɛ\overset{\Delta}{=}{\frac{\Delta\; f}{1/{NT}} = {\Delta\;{fNT}}}} & (4)\end{matrix}$Using the relative frequency offset ε, the received symbol y_(m)[n] isexpressed as

$\begin{matrix}{{{y_{m}\lbrack n\rbrack} = {{{\mathbb{e}}^{j\frac{2{\pi ɛ}\; n}{N}}{\mathbb{e}}^{{j2\pi ɛ}\;{m{({1 + \alpha})}}}{{\mathbb{e}}^{{j\theta}_{0}}\left( {{h_{m}\lbrack n\rbrack}*{x_{m}\lbrack n\rbrack}} \right)}} + {z_{m}\lbrack n\rbrack}}},} & (5)\end{matrix}$where

$\alpha = \frac{N_{g}}{N}$and

${z_{m}\lbrack n\rbrack} = {{\mathbb{e}}^{j\frac{2{\pi ɛ}\; n}{N}}{\mathbb{e}}^{{j2\pi ɛ}\;{m{({1 + \alpha})}}}{\mathbb{e}}^{j\;\theta_{0}}{{{\overset{\sim}{z}}_{m}\lbrack n\rbrack}.}}$The noise z_(m)[n] is a zero-mean complex-Gaussian random-variable withvariance σ_(Z) ²=σ_({tilde over (Z)}) ² and is independent of thetransmit signal and the channel. To simplify the notation, c_(m)[n] isdefined as

$\begin{matrix}{{c_{m}\lbrack n\rbrack}\overset{\Delta}{=}{\frac{1}{N}{\mathbb{e}}^{{j2\pi ɛ}\;{n/N}}{\mathbb{e}}^{{j2\pi ɛ}\;{m{({1 + \alpha})}}}{{\mathbb{e}}^{{j\theta}_{0}}.}}} & (6)\end{matrix}$The received sample y_(m)[n] is theny _(m) [n]=NC _(m) [n](h _(m) [n]*x _(m) [n])+z _(m) [n]  (7)

At the receiver, the discrete Fourier transform (DFT) is performed onthe received samples y_(m)[n] The DFT of y_(m)[n] in the presence of thecarrier frequency offset ε for

${ɛ} < \frac{1}{2}$isY _(m) [k]=C _(m)[0]H _(m) [k]X _(m) [k]+I _(m) [k]+Z _(m) [k],  (8)where

$\begin{matrix}{{{I_{m}\lbrack k\rbrack} = {\sum\limits_{r = 1}^{N - 1}\;{{C_{m}\lbrack r\rbrack}{X_{m}\left\lbrack {k - r} \right\rbrack}}}},} & (9)\end{matrix}$and C_(m)[k], H_(m)[k], and Z_(m)[k] are the DFTs of c_(m)[n], h_(m)[n],and z_(m)[n], respectively. Using (6), it can be shown that

$\begin{matrix}{{C_{m}\lbrack k\rbrack} = {\left( {\frac{\sin\left( {\pi\left( {ɛ - k} \right)} \right)}{N\;{\sin\left( {{\pi\left( {ɛ - k} \right)}/N} \right)}}{\mathbb{e}}^{{{j\pi}{({ɛ - k})}}{({1 - {1/N}})}}} \right) \cdot {{\mathbb{e}}^{j{\lbrack{{2{\pi ɛ}\;{m{({1 + \alpha})}}} + \theta_{0}}\rbrack}}.}}} & (10)\end{matrix}$In (9), it was assumed that H_(m)[k] and X_(m)[k] are periodic withperiod N to simplify the notation. When the frequency offset ε has amagnitude larger than ½, the frequency offset introduces a cyclic shiftof Y_(m)[k] However, it is assumed herein that

${ɛ} < {\frac{1}{2}.}$When

${{ɛ}\underset{\_}{>}\frac{1}{2}},$other integer frequency offset correction algorithms can be employed.

The cyclic prefix-based (CPB) algorithm and the pilot tone-aided (PTA)algorithm are explained below. The CFO estimation algorithm according tothe present invention overcomes the shortcomings of the CPB and the PTAalgorithms. In one cyclic-prefix-based algorithm, the maximum-likelihood(ML) frequency offset estimator that uses the cyclic prefix samples wasderived for the AWGN channel. The received sample y_(m)[n] for the AWGNchannel is

$\begin{matrix}{{y_{m}\lbrack n\rbrack} = {{{\mathbb{e}}^{j\frac{2{\pi ɛ}\; n}{N}}{\mathbb{e}}^{{j2\pi ɛ}\;{m{({1 + \alpha})}}}{\mathbb{e}}^{{j\theta}_{0}}{x_{m}\lbrack n\rbrack}} + {{z_{m}\lbrack n\rbrack}.}}} & (11)\end{matrix}$If x_(m)[n] is a cyclic prefix sample, x_(m)[n+N] is exactly the same asx_(m)[n]. Then it can be easily seen that y_(m)*[n]y_(m)[n+N] isapproximately equal to e^(j2πε)|x_(m)[n]|² for low noise levels. As usedherein, the “*” is used to signify a complex conjugate. For example,y*_(m)[n] is a complex conjugate of y_(m)[n]. Thus, by measuring thephase of y_(m)*[n]y_(m)[n+N], the frequency offset can be estimated. Thefollowing estimator was formally derived and shown to be the MLestimator for the AWGN channel:

$\begin{matrix}{{\hat{ɛ}}_{CPB} = {\frac{1}{2\pi}{{\measuredangle\left( {\sum\limits_{n = 0}^{N_{g} - 1}\;{{y_{m}^{*}\lbrack n\rbrack}{y_{m}\left\lbrack {n + N} \right\rbrack}}} \right)}.}}} & (12)\end{matrix}$Although the ML estimator performs well for the AWGN channel, itsperformance in frequency selective channels is not satisfactory.Moreover, the performance of this estimator degrades if there is anerror in the symbol timing estimation.

In a pilot-tone-aided (PTA) algorithm, the CPB CFO estimator in (12)uses the time-domain samples y_(m)[n], whereas the PTA CFO estimatoruses the DFT value, Y_(m)[k], of the time-domain samples. The PTA CFOwas developed based on the observation that a channel at each subcarrierdoes not change significantly over two consecutive OFDM symbols.Multiplying Y_(m-1)*[k] by Y_(m)[k] results in

$\begin{matrix}{\;{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} = {{{\mathbb{e}}^{{j2\pi ɛ}\;{({1 + \alpha})}}{{{C_{m - 1}^{*}\lbrack 0\rbrack}{C_{m}\lbrack 0\rbrack}}}{H_{m - 1}^{*}\lbrack k\rbrack}{{H_{m}\lbrack k\rbrack} \cdot {X_{m - 1}^{*}\lbrack k\rbrack}}{X_{m}\lbrack k\rbrack}} + {I_{m}^{\prime}\lbrack k\rbrack} + {Z_{m}^{\prime}\lbrack k\rbrack}}},}} & (13)\end{matrix}$where the inter-carrier interference (ICI) I′_(m)[k] isI′ _(m) [k]=C _(m-1)*[0]H _(m-1) *[k]X _(m-1) *[k]I _(m) [k]+C_(m)[0]H_(m)[k]X_(m)[k]I_(m-1)*[k]+I_(m-1)*[k]I_(m)[k],  (14)and the noise Z′_(m)[k] isZ′ _(m) [k]=(Y _(m-1) [k]−Z _(m-1) [k])*Z _(m) [k]+(Y_(m)[k]−Z_(m)[k])Z_(m-1)*[k]+Z_(m-1)*[k]Z_(m)[k].  (15)

In (13), the fact that the phase of C_(m-1)*[0]C_(m)[0] is equal to2πε(1+α) was used, which can be shown from (10). From (13), it can beseen that the phase of Y_(m-1)*[k]Y_(m)[k] is approximately equal to thephase of X_(m-1)*[k]X_(m)[k] plus 2πε(1+α) if the phase of the channeldoes not change substantially over two OFDM symbols, i.e.,

H_(m)[k]≈

H_(m-1)[k] for all m. Since the transmit symbols X_(m-1)*[k] andX_(m)[k] are known to the receiver for pilot subcarriers, the receivercan estimate the frequency offset ε by measuring the phase ofY_(m-1)*[k]Y_(m)[k]X_(m-1)[k]X_(m)*[k]. The PTA CFO estimator can beexpressed as follows:

$\begin{matrix}{{{\hat{ɛ}}_{PTA} = {\frac{1}{2{\pi\left( {1 + \alpha} \right)}}{\measuredangle\left( {\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}} \right)}}},} & (16)\end{matrix}$where P is the set of pilot subcarriers. This PTA CFO estimator wasshown to perform well in frequency selective channels. Although thisestimator performs well, no optimality is associated with this estimatorunlike the CPB estimator. The PTA CFO estimator can be shown to beoptimal in the maximum-likelihood sense if I_(m)[k] is Gaussian andindependent from I_(m)[l] for l≠k. However, I_(m)[k] is not independentfrom I_(m)[l] for l≠k.

Although the PTA CFO estimator described above performs well infrequency selective channels, it requires pilot subcarriers, whichoccupy valuable bandwidth. The pilot subcarriers can be removed by usingthe M-th power method when M-ary phase shift keying (PSK) or amplitudephase shift keying (APSK) is used for modulation of each subcarrier.Since (X_(m-1)*[k]X_(m)[k])^(M)=1 for M-ary PSK or APSK, the phase of(Y_(m-1)*[k]Y_(m)[k])^(M) is approximately equal to 2πMε(1+α) for lownoise and ICI. Thus, the following estimator can be developed:

$\begin{matrix}{{\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\measuredangle\left( {\sum\limits_{k = 0}^{N - 1}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} \right)}}},} & (17)\end{matrix}$where0≦λ≦M.  (18)The denominator in (17) normalizes the magnitude of(Y_(m-1)*[k]Y_(m)[k])^(M) without altering the phase of(Y_(m-1)*[k]Y_(m)[k])^(M) before summing (Y_(m-1)*[k]Y_(m)[k])^(M) overall subcarriers. Although the similarity of (17) with (16) implies thatλ should be M−1, it is not easy to show what value λ should take for thebest performance of the blind estimator. Thus, the performance of theestimator for different λ is shown by simulation in the next section.

Because of the ambiguity in the phase measurement, the blind CFOestimator performs well only when

${{ɛ} < ɛ_{\max}}\overset{\Delta}{=}{\frac{1}{2{M\left( {1 + \alpha} \right)}}.}$For example, for quadrature phase shift keying (QPSK), ε_(max) isapproximately 0.12 for α=0.05. Theoretically, the acquisition range ofthe blind estimator is M times smaller than that of the PTA estimator.However, in practice, the acquisition range of the PTA estimator is alsolimited by the ICI caused by the CFO because the ICI is significant forlarge values of ε. Thus, the acquisition range of the blind CFOestimator is larger than

$\frac{1}{M}$times of that of the PTA estimator. Although the acquisition range ofthe blind CFO estimator is limited to

$\left( {{- \frac{1}{2{M\left( {1 + \alpha} \right)}}},\frac{1}{2{M\left( {1 + \alpha} \right)}}} \right),$it does not pose a significant problem for small M if the blind CFOestimator is used for CFO tracking.

Unlike the CPB estimator, the performance of the blind CFO estimatordepends on the magnitude of the CFO. The power of the ICI I′_(m)[k] is amonotonically increasing function of the magnitude of the CFO. Thus, theperformance of the blind CFO estimator is better for small frequencyoffsets than for large frequency offsets.

The performance of the blind CFO estimator can be improved by iteratingthe estimation and correction steps because the estimator has improvedperformance for small frequency offsets. Once the CFO is estimated usingthe estimator (17), the CFO can be corrected in the time domain bymultiplying the received signal y_(m)[n] withe^(−j2π{circumflex over (ε)}n/N). If |ε−{circumflex over (ε)}| issmaller than |ε|, then the residual frequency offset ε−{circumflex over(ε)}| causes less ICI than the original frequency offset E. Because ofthis reduced ICI, the estimate of the residual frequency offset willbecome even more accurate if the residual CFO is estimated again with(17). The iteration may be performed on the same received signal or asubsequent received signal. Thus, increasingly accurate frequency offsetestimation and correction can be achieved by iterating the estimationand correction steps.

Since the averaging operation improves the performance of theestimators, the proposed estimator has an advantage over the MLestimator. Moreover, the proposed estimator is not affectedsignificantly by the frequency selectivity of the channel. On the otherhand, the ML estimator is adversely affected by the channel frequencyselectivity.

The performance of the blind CFO estimator is evaluated below bysimulation and is compared to the performance of the CPB estimator andthe PTA estimator. The simulation parameters were chosen as follows. Thenumber of data samples, N, is 1024, and the number of cyclic prefixsamples, N_(g), is 56, resulting in α=7/128. The frequency offset ε isassumed to be uniformly distributed between −0.1 and 0.1. The channelused in this section is either the AWGN channel or a multipath channelwith an exponentially decaying power-delay profile. For the multipathchannel, it was assumed that the channel is stationary over two OFDMsymbols, but the root-mean square (rms) delay spread was chosen to be

$\frac{1}{8}N_{g}$in terms of the number of samples.

Referring now to FIG. 2, the mean-square error (MSE) of the blind CFOestimator for the AWGN channel is shown for different values of λ in(17) when QPSK is employed for each subcarrier. Although the performanceof the blind CFO estimator is not very sensitive to the values of λ, thebest performance can be achieved with λ=M−1=3.

Referring now to FIG. 3, the MSE of the blind CFO estimator is shown forthe multipath channel described earlier. In this case, the value of λsignificantly affects the performance of the blind CFO estimator. As inthe case of the AWGN channel, the performance of the blind CFO estimatoris best when λ is equal to 3, although λ=4 results in better performancethan λ=3 for high SNR. The fact that the estimator with λ=4 performsbetter than λ=3 for high SNR can be explained by noting that the ICI isdominant rather than the noise for high SNR.

Referring now to FIG. 4, the MSE of the blind CFO estimator is shown fora different number of iterations for the estimation and correction stepswith λ=M−1=3. By iterating the estimation and correction steps twice,the performance of the blind CFO estimator improves as is shown.However, additional iteration does not improve the performancesignificantly. This can be explained by the fact that the correctionstep reduces only the ICI, not the noise. In other words, theconvergence shown in FIG. 4 occurs because of the background noise thatcannot be reduced.

Referring now to FIGS. 5 and 6, the performance of the blind CFOestimator is compared to that of the CPB and PTA CFO estimators. Here,for fair comparison, the blind CFO estimator estimates the CFO only oncerather than iterating the estimation and correction multiple times. TheMSEs of the blind, CPB, and PTA estimators are shown in FIG. 5. In caseof the PTA CFO estimator, the MSE for 1024, 256, and 64 pilotsubcarriers were plotted. The 1024, 256, and 64 pilot subcarriers occupy100%, 25%, and 6.25% of the bandwidth, respectively. As can be seen, theCPB CFO estimator suffers from the frequency selectivity of the channel.The blind CFO estimator does not perform as well as the PTA CFOestimator with 1024 pilot subcarriers but outperforms the PTA estimatorwith 256 and 64 subcarriers. This means that at least 25% of thebandwidth can be saved by choosing the blind CFO estimator instead ofthe PTA CFO estimator.

Referring now to FIG. 6, a plot of the BER for time-domain differentialdemodulation with differential QPSK (DQPSK) is shown. As can be seen,the blind CFO estimator is able to follow the BER curve of the systemswithout frequency offset, whereas the other estimators suffer fromperformance degradation except for the PTA CFO estimator with 100% pilotsubcarriers.

The blind CFO estimator according to the present invention is suitablefor OFDM transmission over multipath channels. In contrast to the CPBCFO estimator, the blind CFO estimator does not suffer from thefrequency selectivity of a channel. Unlike the PTA CFO estimator, theblind CFO estimator does not require any pilot tones. Although theacquisition range of the blind CFO estimator is limited to smallfrequency offsets, it can be used for carrier frequency tracking. Theperformance of the blind CFO estimator was evaluated by simulation: itoutperforms the CPB CFO estimator and the PTA CFO estimator even whenthe pilot subcarriers occupy 25% of the total subcarriers. Finally, theperformance of the blind CFO estimator can be improved by iterating theestimation and correction steps since the blind CFO estimator performsbetter for small frequency offsets.

Referring now to FIG. 7, a transceiver 50 according to the presentinvention is shown to include a transmitter 54 and a receiver 58. Thetransmitter 54 receives data, performs coding, multiplexing and/or othertransmitter functions and outputs the data to a communications channelas shown. While a wireless application is shown, skilled artisans willappreciate that OFDM also may be used for wired applications. Thereceiver 58 includes a receiver functions module 62 that performsreceiver functions such as decoding, demultiplexing and other receiverfunctions and outputs received data as shown.

A CFO estimator module 66 communicates with the receiver functionsmodule 62. The CFO estimator estimates CFO based on the CFO estimationdescribed above and below. In some embodiments, the estimation is blindand/or an iterative approach is used. In other embodiments, pilotsubcarriers are used, pilot subcarriers with known pilot symbols and/ora single iteration is performed. A CFO correction module 70 receives theCFO and corrects the received signal. In some embodiments, the receivedsignal in the time domain y_(m)[n] is multiplied bye^(−j2π{circumflex over (ε)}n/N).

Referring now to FIG. 8, steps performed by the blind CFO estimator andcorrection modules are shown. In step 100, X is set equal to 1. In step104, the DFT of received signals for adjacent symbols is performed. Instep 108, the DFT of subcarriers for adjacent symbols is multiplied togenerate a product. In step 112, the M^(th) power of the product iscalculated to generate a first value. In step 116, the λ^(th) power ofthe magnitude of the product is calculated to generate a normalizingvalue. In step 120, the first value is divided by the normalizing valueto generate a normalized value. In step 124, the normalized values aresummed over all of the subcarriers and the phase of the sum isdetermined. In step 132, the phase is multiplied by

$\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}$to generate the CFO {circumflex over (ε)}.

CFO correction can then be performed. In step 140, the received signalin the time domain y_(m)[n] is multiplied bye^(−j2π{circumflex over (ε)}n/N) to adjust the received signal{circumflex over (ε)}. In step 144, control determines whether X=Y,where Y is the number of iterations to be performed. Y can be set equalto any integer, although diminishing improvement may occur as Yincreases. In some embodiments, Y=2. If step 144 is false, X isincremented in step 146 and control returns to step 104. Otherwise,control ends in step 148. Steps 100, 144 and 146 can be omitted if asingle iteration is to be performed.

In some implementations, the CFO is used to adjust a digital clock inthe system that generates the carrier frequency rather than adjustingthe analog system clock. The coarse estimate can be generated by anotheralgorithm such as the CPB algorithm, although other algorithms can beused.

The previous description related to a blind carrier frequency trackingalgorithm and demonstrated that the blind algorithm works very well inthe absence of the pilots. It was shown by simulation that the blindalgorithm outperforms the pilot-tone aided (PTA) algorithm, which thatrequires the pilot subcarriers, when the number of pilot subcarriers isless than 25% of the total subcarriers. However, many of the currentsystems already have pilot subcarriers. For example, HD radio anddigital video broadcasting (DVB) systems employ pilot subcarriers. Theblind algorithm can be modified according to the present invention forsystems with pilot subcarriers.

In some implementations according to the invention, the pilotsubcarriers are treated in the same way as data subcarriers. Thus, theestimator becomes:

${\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {\sum\limits_{{k\;\varepsilon\; P}\bigcup D}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} \right)}}},$where D is a set of data subcarrier indices and P is a set of pilotsubcarrier indices. However, this estimator does not exploit the factthat the pilot subcarriers contain known symbols.

An improved estimator according to some implementations of the presentinvention that uses the known pilot symbols is:

$\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {{\sum\limits_{k\;\varepsilon\; D}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} + \frac{\left( {\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}} \right)^{M}}{{{\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}}^{\lambda}}} \right)}}$

Usually, taking the M-th power increases the noise power. The firstestimator takes the M-th power for each subcarrier and sums the M-thpowered values, while the second estimator takes the M-th power aftersumming the values over all pilot subcarriers. Thus, the secondestimator has smaller noise than the first estimator, which providesimproved performance with respect to the first estimator.

As compared to the existing PTA estimator described above, theacquisition range of the above two estimators can be smaller. However,the MSE of this estimator will be much smaller than the PTA estimator.Thus, depending on the applications, the proposed estimators can be moreuseful than the PTA estimator.

Those skilled in the art can now appreciate from the foregoingdescription that the broad teachings of the present invention can beimplemented in a variety of forms. Therefore, while this invention hasbeen described in connection with particular examples thereof, the truescope of the invention should not be so limited since othermodifications will become apparent to the skilled practitioner upon astudy of the drawings, the specification and the following claims.

1. A system comprising: a receiver module that generates a receivercarrier frequency and that demodulates an orthogonal frequency divisionmultiplexing (OFDM) signal using said receiver carrier frequency; and aCFO estimator module that communicates with said receiver module andthat generates a carrier frequency offset (CFO) estimate of said OFDMsignal using a method that includes an exponential product based onadjacent symbols and a normalized magnitude, wherein said CFO estimatormodule generates said CFO estimate based on${\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {\overset{N - 1}{\sum\limits_{k\; = \; 0}}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} \right)}}},$wherein M is a number of phases, λ is a normalizing power, N is a numberof data samples, α is a ratio of a number of cyclic prefix samplesdivided by N, m is a symbol index, and Y_(m-1)*[k]Y_(m)[k] are adjacentreceived symbols.
 2. The system of claim 1 wherein said OFDM signalcontains pilot subcarriers.
 3. The system of claim 2 wherein said pilotsubcarriers contain known pilot symbols.
 4. The system of claim 1wherein said receiver module operates in a frequency-selective channel.5. The system of claim 1 wherein said receiver module operates in anadditive white Gaussian noise (AWGN) channel.
 6. The system of claim 1wherein said receiver module operates in a multi-path fading channel. 7.The system of claim 1 wherein said receiver module is a wired receiver.8. The system of claim 1 wherein said receiver module is a wirelessreceiver.
 9. The system of claim 1 wherein said receiver module performsdemodulation using amplitude phase shift keying (APSK).
 10. The systemof claim 1 wherein said receiver module performs demodulation usingM-ary phase shift keying (MPSK).
 11. The system of claim 1 wherein saidCFO estimator module generates a first CFO estimate_(ɛ̂_1), multiplies atime domain signal by 𝕖^(−j2πɛ̂_(_1)n/N) to generate an adjusted timedomain signal, and uses the adjusted time domain signal to generate asecond CFO estimate estimate_(ɛ̂_2), where n and N are integers.
 12. Thesystem of claim 11 wherein said CFO estimator module multiplies saidadjusted time domain signal by 𝕖^(−j2πɛ̂_(_2)n/N) to generate a secondadjusted time domain signal, and uses the second adjusted time domainsignal to generate a third CFO estimate_(ɛ̂_3).
 13. The system of claim 1wherein 0≦λ≦M.
 14. A system comprising: receiver means for generating areceiver carrier frequency and for demodulating an orthogonal frequencydivision multiplexing (OFDM) signal using said receiver carrierfrequency; and estimator means, that communicates with said receivermeans, for generating a carrier frequency offset (CFO) estimate of saidOFDM signal using a method that includes an exponential product based onadjacent symbols and a normalized magnitude, wherein said estimatormeans generates said CFO estimate based on${\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {\overset{N - 1}{\sum\limits_{k\; = \; 0}}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} \right)}}},$wherein M is a number of phases, λ is a normalizing power, N is a numberof data samples, α is a ratio of a number of cyclic prefix samplesdivided by N, m is a symbol index, and Y_(m-1)*[k]Y_(m)[k] are adjacentreceived symbols.
 15. The system of claim 14 wherein said OFDM signalcontains pilot subcarriers.
 16. The system of claim 14 wherein saidpilot subcarriers contain known pilot symbols.
 17. The system of claim14 wherein said receiver means operates in a frequency-selectivechannel.
 18. The system of claim 14 wherein said receiver means operatesin an additive white Gaussian noise (AWGN) channel.
 19. The system ofclaim 14 wherein said receiver means operates in a multi-path fadingchannel.
 20. The system of claim 14 wherein said receiver means is awired receiver.
 21. The system of claim 14 wherein said receiver meansis a wireless receiver.
 22. The system of claim 14 wherein said receivermeans performs demodulation using amplitude phase shift keying (APSK).23. The system of claim 14 wherein said receiver means performsdemodulation using M-ary phase shift keying (MPSK).
 24. The system ofclaim 14 wherein said estimator means generates a first CFO${estimate}_{\overset{\bigwedge}{ɛ}\_ 1},$ multiplies a time domainsignal by${\mathbb{e}}^{{- j}\; 2\pi\;{\overset{\bigwedge}{ɛ}}_{\_ 1}{n/N}}$ togenerate an adjusted time domain signal, and uses the adjusted timedomain signal to generate a second CFO${estimate}_{\overset{\bigwedge}{ɛ}\_ 2},$ where n and N are integers.25. The system of claim 24 wherein said estimator means multiplies saidadjusted time domain signal by${\mathbb{e}}^{{- j}\; 2\pi\;{\overset{\bigwedge}{ɛ}}_{\_ 2}{n/N}}$ togenerate a second adjusted time domain signal, and uses the secondadjusted time domain signal to generate a third CFO${estimate}_{\overset{\bigwedge}{ɛ}\_ 3}.$
 26. The system of claim 14wherein 0≦λ≦M.
 27. A method comprising: generating a receiver carrierfrequency; demodulating an orthogonal frequency division multiplexing(OFDM) signal using said receiver carrier frequency in a receiver;generating a carrier frequency offset (CFO) estimate of said OFDM signalusing a method that includes an exponential product based on adjacentsymbols and a normalized magnitude, where generating said CFO estimateis based on${\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {\overset{N - 1}{\sum\limits_{k\; = \; 0}}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} \right)}}},$wherein M is a number of phases, λ is a normalizing power, N is a numberof data samples, α is a ratio of a number of cyclic prefix samplesdivided by N, m is a symbol index, and Y_(m-1)*[k]Y_(m)[k] are adjacentreceived symbols.
 28. The system of claim 27 wherein said OFDM signalcontains pilot subcarriers.
 29. The system of claim 27 wherein saidpilot subcarriers contain known pilot symbols.
 30. The method of claim27 further comprising operating in a frequency-selective channel. 31.The method of claim 27 further comprising operating in an additive whiteGaussian noise (AWGN) channel.
 32. The method of claim 27 furthercomprising operating in a multi-path fading channel.
 33. The method ofclaim 27 further comprising performing demodulation using amplitudephase shift keying (APSK).
 34. The method of claim 27 further comprisingperforming demodulation using M-ary phase shift keying (MPSK).
 35. Themethod of claim 27 further comprising: generating a first CFO${estimate}_{\overset{\bigwedge}{ɛ}\_ 1};$ multiplying a time domainsignal by${\mathbb{e}}^{{- j}\; 2\pi\;{\overset{\bigwedge}{ɛ}}_{\_ 1}{n/N}}$ togenerate an adjusted time domain signal, where n and N are integers; andusing the adjusted time domain signal to generate a second CFO${estimate}_{\overset{\bigwedge}{ɛ}\_ 2}.$
 36. The method of claim 35further comprising: multiplying said adjusted time domain signal by${\mathbb{e}}^{{- j}\; 2\pi\;{\overset{\bigwedge}{ɛ}}_{\_ 2}{n/N}}$ togenerate a second adjusted time domain signal; and using the secondadjusted time domain signal to generate a third CFO estimate  ɛ̂_(_3).37. The method of claim 27 wherein 0≦λ≦M.
 38. A system comprising: areceiver module that generates a receiver carrier frequency and thatdemodulates an orthogonal frequency division multiplexing (OFDM) signalusing said receiver carrier frequency; and a CFO estimator module thatcommunicates with said receiver module and that estimates carrierfrequency offset (CFO) of said OFDM signal using a method that includesan exponential product based on adjacent symbols and a normalizedmagnitude, wherein said OFDM signal contains pilot subcarriers, andwherein said CFO estimator module estimates said CFO based on:$\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {{\sum\limits_{k\;\varepsilon\; D}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} + \frac{\left( {\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}} \right)^{M}}{{{\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}}^{\lambda}}} \right)}}$wherein M is a number of phases, λ is a normalizing power, N is a numberof data samples, α is a ratio of a number of cyclic prefix samplesdivided by N, m is a symbol index, Y_(m-1)*[k]Y_(m)[k] are adjacentreceived symbols, X_(m-1)*[k]X_(m)[k] are pilot subcarriers, D is a setof data subcarrier indices, and P is a set of pilot subcarrier indices.39. The system of claim 38 wherein said pilot subcarriers contain knownpilot symbols.
 40. The system of claim 38 wherein said receiver moduleoperates in a frequency-selective channel.
 41. The system of claim 38wherein said receiver module operates in an additive white Gaussiannoise (AWGN) channel.
 42. The system of claim 38 wherein said receivermodule operates in a multi-path fading channel.
 43. The system of claim38 wherein said receiver module is a wired receiver.
 44. The system ofclaim 38 wherein said receiver module is a wireless receiver.
 45. Thesystem of claim 38 wherein said receiver module performs demodulationusing amplitude phase shift keying (APSK).
 46. The system of claim 38wherein said receiver module performs demodulation using M-ary phaseshift keying (MPSK).
 47. The system of claim 38 wherein said CFOestimator module generates a first CFO estimate_(ɛ̂_1), multiplies a timedomain signal by 𝕖^(−j2πɛ̂_(_1)n/N) to generate an adjusted time domainsignal, and uses the adjusted time domain signal to generate a secondCFO estimate_(ɛ̂_2), where n and N are integers.
 48. The system of claim47 wherein said CFO estimator module multiplies said adjusted timedomain signal by 𝕖^(−j2πɛ̂_(_2)n/N) to generate a second adjusted timedomain signal, and uses the second adjusted time domain signal togenerate a third CFO estimate  ɛ̂_(_3).
 49. The system of claim 38wherein 0≦λ≦M.
 50. A system comprising: receiver means for generating areceiver carrier frequency and for demodulating an orthogonal frequencydivision multiplexing (OFDM) signal using said receiver carrierfrequency; and estimator means, that communicates with said receivermeans, for generating a carrier frequency offset (CFO) estimate of saidOFDM signal using a method that includes an exponential product based onadjacent symbols and a normalized magnitude, wherein said OFDM signalcontains pilot subcarriers, and wherein said estimator means generatessaid CFO estimate based on:$\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {{\sum\limits_{k\;\varepsilon\; D}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} + \frac{\left( {\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}} \right)^{M}}{{{\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}}^{\lambda}}} \right)}}$wherein M is a number of phases, λ is a normalizing power, N is a numberof data samples, α is a ratio of a number of cyclic prefix samplesdivided by N, m is a symbol index, Y_(m-1)*[k]Y_(m)[k] are adjacentreceived symbols, X_(m-1)*[k]X_(m)[k] are pilot subcarriers, D is a setof data subcarrier indices, and P is a set of pilot subcarrier indices.51. The system of claim 50 wherein said pilot subcarriers contain knownpilot symbols.
 52. The system of claim 50 wherein said receiver meansoperates in a frequency-selective channel.
 53. The system of claim 50wherein said receiver means operates in an additive white Gaussian noise(AWGN) channel.
 54. The system of claim 50 wherein said receiver meansoperates in a multi-path fading channel.
 55. The system of claim 50wherein said receiver means is a wired receiver.
 56. The system of claim50 wherein said receiver means is a wireless receiver.
 57. The system ofclaim 50 wherein said receiver means performs demodulation usingamplitude phase shift keying (APSK).
 58. The system of claim 50 whereinsaid receiver means performs demodulation using M-ary phase shift keying(MPSK).
 59. The system of claim 50 wherein said estimator meansgenerates a first CFO estimate_(ɛ̂_1), multiplies a time domain signal by𝕖^(−j2πɛ̂_(_1)n/N) to generate an adjusted time domain signal, and usesthe adjusted time domain signal to generate a second CFOestimate  ɛ̂_(_2), where n and N are integers.
 60. The system of claim 59wherein said estimator means multiplies said adjusted time domain signalby 𝕖^(−j2πɛ̂_(_2)n/N) to generate a second adjusted time domain signal,and uses the second adjusted time domain signal to generate a third CFOestimate  ɛ̂_(_3).
 61. The system of claim 50 wherein 0≦λ≦M.
 62. A methodcomprising: generating a receiver carrier frequency; demodulating anorthogonal frequency division multiplexing (OFDM) signal using saidreceiver carrier frequency in a receiver; and estimating carrierfrequency offset (CFO) of said OFDM signal using a method that includesan exponential product based on adjacent symbols and a normalizedmagnitude, wherein estimating said CFO is based on:$\hat{ɛ} = {\frac{1}{2\pi\;{M\left( {1 + \alpha} \right)}}{\angle\left( {{\sum\limits_{k\;\varepsilon\; D}\;\frac{\left( {{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}} \right)^{M}}{{{{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}^{\lambda}}} + \frac{\left( {\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}} \right)^{M}}{{{\sum\limits_{k\;\varepsilon\; P}\;{{X_{m - 1}\lbrack k\rbrack}{X_{m}^{*}\lbrack k\rbrack}{Y_{m - 1}^{*}\lbrack k\rbrack}{Y_{m}\lbrack k\rbrack}}}}^{\lambda}}} \right)}}$wherein M is a number of phases, λ is a normalizing power, N is a numberof data samples, α is a ratio of a number of cyclic prefix samplesdivided by N, m is a symbol index, Y_(m-1)*[k]Y_(m)[k] are adjacentreceived symbols X_(m-1)*[k]X_(m)[k] are pilot subcarriers, D is a setof data subcarrier indices, and P is a set of pilot subcarrier indices,wherein said OFDM signal contains pilot subcarriers.
 63. The method ofclaim 62 wherein said pilot subcarriers contain known pilot symbols. 64.The method of claim 62 further comprising operating in afrequency-selective channel.
 65. The method of claim 62 furthercomprising operating in an additive white Gaussian noise (AWGN) channel.66. The method of claim 62 further comprising operating in a multi-pathfading channel.
 67. The method of claim 62 further comprising performingdemodulation using amplitude phase shift keying (APSK).
 68. The methodof claim 62 further comprising performing demodulation using M-ary phaseshift keying (MPSK).
 69. The method of claim 62 further comprising:generating a first CFO estimate_(ɛ̂_1); multiplying a time domain signalby 𝕖^(−j2πɛ̂_(_1)n/N) to generate an adjusted time domain signal, where nand N are integers; and using the adjusted time domain signal togenerate a second CFO estimate estimate  ɛ̂_(_2).
 70. The method of claim69 further comprising: multiplying said adjusted time domain signal by𝕖^(−j2πɛ̂_(_2)n/N) to generate a second adjusted time domain signal; andusing the second adjusted time domain signal to generate a third CFOestimates estimate  ɛ̂_(_3).
 71. The method of claim 62 wherein 0≦λ≦M.72. The system of claim 1 wherein said exponential product includesmultiplication of a discrete Fourier transform of adjacent symbols ofsaid OFDM signal.
 73. The system of claim 1 wherein said CFO estimatorgenerates said CFO estimate of said OFDM signal based on saidexponential product and a normalized magnitude of that product.